Get yourself s multimeter and use normal troubleshooting procedures. Connect it to the pump wiring as near to the pump as you can. Or better yet if you have a clamp-on DC ammeter. Try to not disturb the wiring any more than absolutely required.
When the pump is on (add water if you need to to turn it on,) monitor the voltage (or current) and look for sagging voltage. If it sags more than a volt or two, you have a corroded connection to find. Visually you may never see it. If full voltage is present the problem is in the pump.
Ken
I do have a clip-on ammeter, and in principle that's a good suggestion. There are two problems with this approach, however: first, I'd have to be staring at it for up to an hour waiting for it to go off, and second the duration is so short (it's usually closer to 2 seconds than 5) it might not even register. Others in this thread have given me an idea for a better way to find it.
I think the consensus among those who carefully read what I wrote is that the problem MUST be in the wiring, and although until now I thought that pretty far-fetched, the more I think about it, the more I realize it is really the only possibility that makes sense. If you consider the system to comprise three components (pump with level switch, Auto/Off/Manual switch, and the wiring) unless I have astoundingly bad luck it can't be the pump (third one) or the Auto/Off/Manual switch (second one), which leaves only the wiring.
And it has to be between the Auto/Off/Manual switch and the pump. Somehow 12V is briefly getting to the manual wire (brown w. white) to the pump, causing it to run. It can't be the auto wire (brown), because it's 12V all the time. Now, it makes sense that the source of the 12V has to be the auto wire shorting to the manual wire. If it was any other source of 12V, the pump would run but the indicator light on the panel wouldn't light.